Koszulity, supersolvability, and Stirling representations

Abstract

Supersolvable hyperplane arrangements and matroids are known to give rise to certain Koszul algebras, namely their Orlik-Solomon algebras and graded Varchenko-Gel'fand algebras. We explore how this interacts with group actions, particularly for the braid arrangement and the action of the symmetric group, where the Hilbert functions of the algebras and their Koszul duals are given by Stirling numbers of the first and second kinds, respectively. The corresponding symmetric group representations exhibit branching rules that interpret Stirling number recurrences, which are shown to apply to all supersolvable arrangements. They also enjoy representation stability properties that follow from Koszul duality.

v3: final version, to appear in Annals of Representation Theory

References in corpus (14)

• FI-modules and stability for representations of symmetric groups
• Representation theory and homological stability
• Representation stability and finite linear groups
• Gröbner-Shirshov bases and their calculation
• The stability of the Kronecker products of Schur functions
• Formality properties of finitely generated groups and Lie algebras
• On the homotopy Lie algebra of an arrangement
• The Varchenko-Gel'fand Ring of a Cone
• On a curious variant of the $S_n$-module $Li{e}_n$
• Eulerian representations for real reflection groups
• The holonomy Lie algebra of a matroid
• The reflection representation in the homology of subword order
• Koszulity of dual braid monoid algebras via cluster complexes
• Equivariant cohomology and conditional oriented matroids